3.3 Black–Scholes–Merton

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma and Multivariable Calculus. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

The Black–Scholes–Merton (BSM) model is the continuous-time limit of binomial dynamic replication, and it is the zero point of all modern derivative pricing. Its claim is sharp: if you can trade the underlying continuously, then the price of a European option is uniquely fixed by the no-arbitrage condition - it depends on the current spot, the strike, the remaining time, the risk-free rate, and the (assumed constant) volatility, and on nothing else. Not on the drift of the stock, not on your forecast of where the market is going.

This folder is the model topic-folder for the Kwant-Atlas build. It is a hub: it (a) gives you the fast formula lookup below (job #1 of this pillar), and (b) routes you to six sub-pages that walk you from raw intuition through the derivations, the closed forms, the Greeks, the failure modes, and the extensions.

The one-sentence essence. “Price a derivative as the discounted expectation of its payoff under the risk-neutral measure - equivalently, solve the BSM parabolic PDE - where the change-of-measure/derivation is driven entirely by volatility, never by drift.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). All formulas below are transcribed from Haug (2006) and cross-checked against Shreve Vol II Ch 4–5; the numbers in the check column were re-executed and reproduced exactly from the verified corpus (see §3).

Notation: $S$ spot, $X$ strike, $T$ time to expiry (years), $r$ risk-free rate, $q$ continuous dividend yield, $b$ cost-of-carry, $σ$ vol, $N (\cdot)$ standard normal CDF, $n (x) = \frac{1}{2 π} e^{- x^{2} /2}$ .

Cost-of-carry dictionary (one master formula, many models): $b = r$ stock · $b = r - q$ stock/index with yield $q$ · $b = 0$ futures (Black-76) · $b = r - r_{f}$ currency (Garman–Kohlhagen).

Quantity	Formula	Verified check
$d_{1}$	$\frac{ln ( S / X ) + ( b + \frac{1}{2} σ ^{2} ) T}{σ T}$	-
$d_{2}$	$d_{1} - σ T = \frac{ln ( S / X ) + ( b - \frac{1}{2} σ ^{2} ) T}{σ T}$	-
Generalized call $c$	$S e^{(b - r) T} N (d_{1}) - X e^{- r T} N (d_{2})$	$S = 60, X = 65, T = .25, r = .08, σ = .30 \Rightarrow c = 2.13337$
Generalized put $p$	$X e^{- r T} N (- d_{2}) - S e^{(b - r) T} N (- d_{1})$	Merton $p = 2.46479$ ( $q = 5%$ )
Put–call parity	$c - p = S e^{(b - r) T} - X e^{- r T}$	$S = 100, X = 105, T = .5, r = .10$ : $C + X e^{- r T} = P + S = 105.57354$
Delta call	$e^{(b - r) T} N (d_{1})$	$0.503105$
Gamma (call=put)	$\frac{e ^{(b - r) T} n ( d _{1} )}{S σ T}$	$0.026794$
Vega (call=put)	$S e^{(b - r) T} n (d_{1}) T$	raw $19.2999$ , per 1 vol-point $0.1930$
Theta call (per day)	$\frac{1}{365} [- \frac{S e ^{(b - r) T} n ( d _{1} ) σ}{2 T} - (b - r) S e^{(b - r) T} N (d_{1}) - r X e^{- r T} N (d_{2})]$	$- 0.036989$ /day
Rho call	$T X e^{- r T} N (d_{2})$	raw $10.9656$ , per 1 rate-point $0.1097$
Gamma–theta trade	$\frac{1}{2} Γ S^{2} σ^{2} = - Θ_{driftless}$	both $= 11.5800$
ATM-forward approx	$c \approx p \approx 0.4 S e^{(b - r) T} σ T$	$3.8713$ vs exact $3.8579$

Critical scaling caveat (Haug §2). Screen/lookup values for Vega, Rho, Phi, Carry, Vanna, Zomma are quoted per 1 vol/rate point = raw $/100$ ; Vomma $/10 000$ ; Ultima $/1 0^{6}$ ; Theta per day = raw $/365$ . Failing to apply this convention is the single most common lookup error.

3. Computational Implementation - the formula engine

This runs on the standard library only (math.erf gives the exact normal CDF, so there is no dependency on numpy/scipy). It reproduces every verified number above.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

Constant-vol delusion - BSM forces one $σ$ ; real markets show a smile, so the only free parameter cannot fit all strikes.
Continuous-hedging friction - the “riskless portfolio” needs $d t \to 0$ ; costs and discrete rebalancing turn it into a loss maker (measured via the gamma–theta residual).
Lognormal/completeness failure - jumps and fat tails break the single-Brownian-motion assumption that makes prices unique (the structural route into Heston/SABR and jump models).

5. References

Haug, Espen Gaarder: The Complete Guide to Option Pricing Formulas (2nd ed., 2006)
Shreve, Steven E.: Stochastic Calculus for Finance II
Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
Björk, Tomas: Arbitrage Theory in Continuous Time (3rd ed.)

6. Connected Graph Bridges

Foundational base: Stochastic Calculus & Itô · Calculus
Sibling topic: No-Arbitrage Foundations & Binomial Trees (the discrete seed of this model)
Sub-pages (in-folder): 01 From Zero · 02 PDE & Derivation · 03 Pricing Formulas · 04 Greeks & Hedging · 05 Failure Modes · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05

Kwant Atlas

Explorer

3.3 Black–Scholes–Merton

1. Intuition & Practical Objective

2. Mathematical Ground Truth & Derivations

3. Computational Implementation - the formula engine

4. Failure Modes & First-Principles Breakdowns

5. References

6. Connected Graph Bridges

3.3.1 Black–Scholes–Merton from Zero

3.3.2 The BSM PDE, Its Derivation & the Feynman–Kac Bridge

3.3.3 The European Pricing Formulas (Closed Forms, Dividends, FX, Futures)

3.3.4 The Greeks & Dynamic Hedging (Full Sensitivity Lookup)

3.3.5 Failure Modes & Real-World Practice

3.3.6 Advanced Extensions